Notes

Outline

NON-LINEAR CONVOLUTION: A NEW APPROACH FOR THE AURALIZATION OF DISTORTING SYSTEMS Angelo Farina, Alberto Bellini and Enrico Armelloni Industrial Engineering Dept., University of Parma, Via delle Scienze 181/A Parma, 43100 ITALY – HTTP://pcfarina.eng.unipr.it

Goals for Auralization Transform the results of objective electroacoustics measurements to audible sound samples suitable for listening tests

Methods We start from a measurement of the system based on exponential sine sweep (Farina, 108th AES, Paris 2000) Diagonal Volterra kernels are obtained by post-processing the measurement results These kernels are employed as FIR filters in a multiple-order convolution process (original signal, its square, its cube, and so on are convolved separately and the result is summed)

Exponential sweep measurement The excitation signal is a sine sweep with constant amplitude and exponentially-increasing frequency

Raw response of the system Many harmonic orders do appear as colour stripes

Deconvolution of system’s impulse response The deconvolution is obtained by convolving the raw response with a suitable inverse filter

Multiple impulse response obtained The last peak is the linear impulse response, the preceding ones are the harmonic distortion orders

Auralization by linear convolution Convolving a suitable sound sample with the linear IR, the frequency response and temporal transient effects of the system can be simulated properly

What’s missing in linear convolution ? No harmonic distortion, nor other nonlinear effects are being reproduced.  From a perceptual point of view, the sound is judged “cold” and “innatural” A comparative test between a strongly nonlinear device and an almost linear one does not reveal any audible difference, because the nonlinear behavior is removed for both

Theory of nonlinear convolution The basic approach is to convolve separately, and then add the result, the linear IR, the second order IR, the third order IR, and so on. Each order IR is convolved with the input signal raised at the corresponding power:

Volterra kernels and simplification The general Volterra series expansion is defined as:

Memoryless distortion followed by a linear system with memory The first nonlinear system is assumed to be memory-less, so only the diagonal terms of the Volterra kernels need to be taken into account.

Volterra kernels from the measurement results The measured multiple IRs h’ can be defined as:

Finding the connection point Going to frequency domain by taking the FFT, the first equation becomes:

Solution Thus we obtain a linear equation system:

Non-linear convolution As we have got the Volterra kernels already in frequency domain, we can efficiently use them in a multiple convolution algorithm implemented by overlap-and-save of the partitioned input signal:

Software implementation Although today the algorithm is working off-line  (as a mix of manual CoolEdit operations and some Matlab processing), a more efficient implementation as a CoolEdit plugin is being worked out:

Audible evaluation of the performance Original signal

Subjective listening test A/B comparison Live recording & non-linear auralization 12 selected subjects 4 music samples 9 questions 5-dots horizontal scale Simple statistical analysis of the results A was the live recording, B was the auralization, but the listener did not know this

Conclusion Statistical parameters – more advanced statistical methods would be advisable for getting more significant results